Let $\Omega$ be a bounded $C^{ 2,\alpha}$ domain in $\R^n$ ($n\geq 1$, $0<\alpha<1$), $\Omega^{ \ast}$ be the open Euclidean ball centered at 0 having the same Lebesgue measure as $\Omega$, $\tau\geq 0$ and $v\in L^{ \infty}(\Omega,\R^n)$ with $\left\Vert v\right\Vert\_{ \infty}\leq \tau$. If $\lambda\_{ 1}(\Omega,\tau)$ denotes the principal eigenvalue of the operator $-\Delta+v\cdot\nabla$ in $\Omega$ with Dirichlet boundary condition, we establish that $\lambda\_{ 1}(\Omega,v)\geq \lambda\_{ 1}(\Omega^{ \ast},\tau e\_{ r})$ where $e\_{ r}(x)=x/| x|$. Moreover, equality holds only when, up to translation, $\Omega=\Omega^{ \ast}$ and $v=\tau e\_{ r}$. This result can be viewed as an isoperimetric inequality for the first eigenvalue of the Dirichlet Laplacian with drift. It generalizes the celebrated Rayleigh-Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian.
Categories: math.AP